Block-Parallel Chaotic Algorithms for Image Reconstruction

Transcription

1 N. Gubaren Poltechna Cz stochowsa ul. Dabrowsego 69, 4- Cz stochowa, Poland, E-al: M. Pleszczyns Poltechna Slasa ul. Kaszubsa, 3, 44-1 Glwce, Poland, E-al: Bloc-Parallel Chaotc Algorths for Iage Reconstructon Recoended by Prof. E. Dshalalow) The paper s devoted to the elaboraton and pleentaton of bloc-parallel asynchronous algorths for coputer toography. The nuercal reconstructon algorths and nuercal sulaton results for a nuber of odelng objects and soe partcular systes of reconstructon are presented. Ðàçðàáîòàíû è âûïîëíåíû áëî íî-ïàðàëëåëüíûå àëãîðèòìû êîìïüþòåðíîé òîìîãðàôèè. Ïðåäñòàâëåíû èñëåííûå àëãîðèòìû âîññòàíîâëåíèÿ è ðåçóëüòàòû èñëåííîãî ìîäåëèðîâàíèÿ äëÿ ðÿäà òåñòîâûõ çàäà è íåêîòîðûõ àñòíûõ ñëó àåâ ñèñòåì ðåêîíñòðóêöèè ñáîðà äàííûõ. K e y w o r d s: coputer toography, ncoplete projecton data, asynchronous algorths, coputer reconstructon, bloc-parallel algorths. Introducton. Soe parallel pleentatons of teratve algebrac algorths for age reconstructon for soe partcular reconstructon schees whch arse n soe probles of engneerng geophyscs and neral ndustry are consdered n the paper. In such coputng structure each eleentary processor executes ts ndependent calculatons by eans of the sae sple algorths connected wth a set of correspondng equatons. It s assued that each processor executes ts calculatons wth ts own pace and the councaton channels are allowed to delver essages out of order. In ths case ths results n the chaotc character of nteractons n such coputer parallel structure CPS) whch corresponds to soe chaotc teratve algorth. Ths algorth realzed on such CPS s based on the asynchronous ethods [1 3]. ISSN Ýëåêòðîí. ìîäåëèðîâàíèå. 9. Ò. 31. ¹ 41

2 N. Gubaren, M. Pleszczyns In order to reduce the coputaton te and eory space of the coputer other algebrac algorths were proposed whch allow ther parallelzaton and ay be realzed on the fast assvely parallel coputng systes MPCS) consstng of eleentary processors and a central processor [4 6]. We represent n ths paper soe nds of the bloc-parallel asynchronous algorths for age reconstructon whch are a certan generalzaton of parallel chaotc teraton ethods consdered by Bru, Elsner and Neuann [7]. Nuercal sulaton of the solvng the probles of age reconstructon fro ncoplete projecton data for soe odelng objects, coparng the errors evaluatons and rate of convergence of these algorths are presented. It s shown that for soe choce of paraeters one can obtan a good qualty of reconstructon wth these algorths, and that these algorths have uch hgher rate of convergence n coparson wth the correspondng synchronous algorths. Bloc-parallel teratve algorths for age reconstructon. Certan parallel and bloc-teratve algorths are used n the paper, soe of whch were consdered n papers [8 ], for solvng the syste of lnear equatons Ax p, 1), where A a j ) R n T n s the atrx of coeffcents; x x1, x,..., x n ) R s T the age vector; p p1, p,..., p ) R s the easureent vector of projecton data; and syste of lnear nequaltes pe Ax p+ e, ) where e { 1,,..., } s a non-negatve vector. Denote by a, x) p ) p a, x)) P x) x a, 3) a where and s s, f s ;, otherwse P 1 ) IP, 4) where a s -th row of a atrx A, s a relaxaton paraeter. Algorth1PART). 1. x ) R n s an arbtrary vector.. The + 1-th teraton s calculated n accordance wth such a schee: y, ) P x 1,,..., ), ) 4 ISSN Electronc Modelng. 9. V. 31. ¹

3 Bloc-Parallel Chaotc Algorths for Iage Reconstructon 1), 1 x C B y, 6) where P are operators defned by 3) and 4), are relaxaton paraeters, C s a constraned operator and B are atrces of denson nn wth real nonnegatve eleents and B = E, B 1, for all N 1 1, where E s the unt atrx of denson n n. The parallel pleentaton of ths algorth ay be organzed as follows: begn x ) =ntal for =, 1,... untl convergence for -th processor, =1to y end P 1) ) x x C B y 1 end end n Let B jj ) j1 be a dagonal atrx wth eleents jj 1.If jj for each j J, I, C = I, then there results the Cno algorth [11]. The suffcent condtons of convergence of algorth 1 are gven by the followng theore. Theore 1. If syste ) s consstent and < <, then the sequence ) x { } 1 defned by algorth 1 converges to soe soluton of the syste ). For any practcal applcatons x, the eleents of a atrx A =a j ) are nonnegatve real nubers and p > for all I. In ths case the followng parallel ultplcatve algorth s proposed for solvng a syste of lnear nequaltes ). Algorth MARTP). 1. x ) R n and x ) >.. The +1-th teraton s calculated n accordance wth such a schee: 1) ), j j j 1 x : x y, 7) ISSN Ýëåêòðîí. ìîäåëèðîâàíèå. 9. Ò. 31. ¹ 43

4 N. Gubaren, M. Pleszczyns where y, j p : ) a, x ) ja j = 1,,..., ; j = 1,,..., n), j are postve real nubers such that 8) a j j 1 1 for every j,. The parallel realzaton of ths algorth ay be gven n such a for: begn x ) j = ntal for =,1,... untl convergence for -th processor, =1to for j-th pont, j =1ton y, j end end p : ) a, x ) x : x y ja j 1) ), j j j 1 1,,..., ) end end The slar algorths for age reconstructon were consdered n wors [11 1]. The suffcent condtons of convergence of the algorth are gven by the followng theore. Theore. Let syste ) be consstent and have f only one postve soluton. If a j, a, p >, j, a j j 1 1 for all, j,, x ) R ) n ), the sequence { x } converges to soe soluton of the syste ). 1 defned by the algorth 44 ISSN Electronc Modelng. 9. V. 31. ¹

5 Bloc-Parallel Chaotc Algorths for Iage Reconstructon The proofs of theores 1 and are presented n [9]. These algorths ay be realzed on parallel coputng structures consstng of eleentary processors and one central processor. On each + 1)-th step of teraton every -th eleentary processor coputes the coordnates of vector y, n accordance wth forula ) or 8) and then the central processor coputes the + 1)-th teraton of the age vector x n accordance wth forula 6) or 7). The an defect of parallel algorths consdered above s ther practcal realzaton on parallel coputatonal structures because t needs a lot of local processors n such MPCS. In order to reduce the nuber of requred local processors we consder a bloc-teratve addtve and ultplcatve algorths. For ths purpose decopose the atrx A and the projecton vector p nto M subsets n accordance wth decoposton {1,,..., }=H 1 H... H M, where Ht { t11, t1,..., t},= < 1 <... < M =. 9) Algorth3. 1. x ) R n s an arbtrary vector.. The +1-th teraton s calculated n accordance wth such a schee: 1 ) Ht) x C B P y, where t )=od M) +1, P are operators defned by 3) and 4), are relaxaton paraeters, C s a constraned operator and B are atrces of denson nnwth real nonnegatve eleents and ) B = E, B 1, Ht) Ht) ffor all N, where E s the unt atrx of the denson n n. The parallel pleentaton of ths algorth can be descrbed as follows: y, ) P x ), x C B y H t or ay be gven by such a for: begn x ) = ntal for =, 1,... untl convergence t)=od M)+1 for -th processor, H t 1), Ht) ISSN Ýëåêòðîí. ìîäåëèðîâàíèå. 9. Ò. 31. ¹ 4,

6 N. Gubaren, M. Pleszczyns y, ) P x end x ) 1 : C B y Ht end end The condtons of convergence of algorth 3 ay be gven by the followng theore. Theore 3. If syste ) s consstent and, then for every pont x ) R n the sequence{ ) x } 1 defned by algorth 3 converges to soe soluton of the syste ). The bloc-teratve algorths represent exaples of sequental-parallel algorths. They ay be consdered as an nteredate verson between sequental algorths and full parallel ones. In each step of teratve process the bloc-teratve algorth uses sultaneously nforaton about all equatons concernng the gven bloc. Bloc-teratve algorths ay be also consdered n the case of ultplcatve algorths. In ths case the followng algorth s obtaned. Algorth4 BMART). 1. x ) R n and x ) >.. The + 1-th teraton s calculated n accordance wth such a schee: where j x 1) ) j : x j Ht) p a, x are postve real nubers such that Ht) a j j 1 ) ja j j, Ht) for every j, ; H t) are defned n accordance wth 9), and t ) s alost cycle control sequence. If for all, j anda j 1, 1, then as a result the bloc-teratve ultplcatve algorth proposed n [6] s obtaned. The condtons of convergence are the sae as for the algorth. Bloc-parallel asynchronous algorths for coputer toography. In ths secton the generalzed odel ofa asynchronous teratons, consdered n [13], s appled for pleentaton of bloc-parallel algorths on nonsynchronous coputer structure. We shall use the basc notons of the theory of asynchronous teratons whch were ntroduced by Chazan and Mraner n [3] and Baudet n [1] see, also [14]). 46 ISSN Electronc Modelng. 9. V. 31. ¹

7 Bloc-Parallel Chaotc Algorths for Iage Reconstructon Applyng the generalzed odel of asynchronous teratons for pleentaton of algorth BPART on nonsynchronous coputer structure results n obtanng the followng algorth, where the nubers of operators are chosen n the chaotc way: Algorth. 1. x ) R n s an arbtrary vector.. The + 1-th teraton s calculated n accordance wth such a schee: Ht) x C B P x where P are operators defned by 3) and 4), are relaxaton paraeters, C s a constraned operator, t )=I, I { I } s a sequence of chaotc sets such that I {, 1,..., } and B are atrces of denson nn wth real nonnegatve eleents whch satsfy the condtons of ), J { )} 1 are se- quences of delays. The convergence of ths algorth s gven by the followng theore. Theore 4. Let syste 1) be consstent, I { I } be a regular sequence of chaotc sets I {, 1,..., } wth the nuber of regularty T, J { )} 1 be sequences wth lted delays and j ) ), and let the nuber of delays be equal to T. Then for every pont x ) R n ) the sequence{ x } 1 defned by algorth converges to soe pont x H, whch s a fxed pont of orthogonal projecton operators P = 1,,..., ). The full proof of ths theore one can fnd n [9]. In ths paper the constraned operator C s gven n the for C = C 1 C, where a, f x a; C1[ x ]) x, f ax b; b, f x b; C [ x]) j, f p and aj ; x j, otherwse. Coputer sulaton and experental results. In ths secton we present soe nuercal results of applyng the specal cases of bloc-parallel algorth BPART-3 and chaotc bloc-parallel algorth CHBP-3 consdered n the prevous sectons for the reconstructon of hgh contrast objects fro ncoplete projecton data n the case when they are not avalable at each angle of vew and they are a few-nuber lted. The nfluence of varous paraeters of these algorths such as a pxel ntalzaton, relaxaton paraeters, the nuber of teratons and nose n the projecton data on reconstructon qualty and convergence of these algorth are also studed there. ISSN Ýëåêòðîí. ìîäåëèðîâàíèå. 9. Ò. 31. ¹ 47,

8 N. Gubaren, M. Pleszczyns Fg. 1. The orgnal functon f x, y) In dependence on obtanng the syste of projectons there are any age reconstructon schees. In soe practcal probles, n engneerng for exaple, t s possble to get projectons fro all drectons because of the exstng of soe portant reasons such as stuaton, sze or possblty of an access to a research object). Ths stuaton arses, for exaple, n the coal bed worng. In ths paper the goodness of the appled algorth of reconstructon was tested for dfferent nds of geoetrc fgures and reconstructon schees. The dscrete functons wth hgh contrast were chosen to llustrate the pleentaton of these algorths worng wth ncoplete data. The results presented n ths paper are gven for the followng functon: 1, x, y) D1 E R ;, x, y) D E R ; f x, y) 3, x, y) D3 E R ; 4, x, y) D E R 4 ;, otherwse, where E s a square E{ x, y): 1x, y 1, } and D are subsets of E of the followng for: D 1 = [.7,.4] [.,.], D = [.,.] [.1,.1], D 3 = [.,.] [.3,.], D 4 = [.4,.7] [.4,.7]. The plot of ths functon s gven n Fg. 1. The results for age reconstructons are represented for only two dfferent schees, whch are descrbed n [14] 48 ISSN Electronc Modelng. 9. V. 31. ¹

9 Bloc-Parallel Chaotc Algorths for Iage Reconstructon Fg.. The schees of obtanng projecton data: syste 1 1) and syste 1 1, 1 1): 1 sources of rays; research object; 3 rays; 4 detectors and were called syste 1 1) and 1 1, 1 1). Both of these systes are shown n Fg.. In the frst schee of obtanng the projecton data, whch we shall call the syste 1 1), we have an access to the research object fro only two opposte sdes. Ths stuaton often arses n engneerng geophyscs. In ths case the sources of rays are located only on one sde and the detectors are located on the opposte sde of the research part of a coal bed. Ths schee of nforaton obtanng s shown n Fg.. The convergence characterstcs of age reconstructon are gven n a vew of plots for the followng easures of errors: ~ the absolute error: x, y ) f x, y ) f x, y) ; ~ the axal absolute error: ax f f ; the axal relatve error: ~ ax f f 1 % ; ax f the ean absolute error: 1 ~ f f, n where f s the value of the gven odelng functon n the center of the -th pxel and f ~ s the value of the reconstructed functon n the -th pxel. As the result of coputer sulaton t was assued, that n s the nuber of pxels,.e. the nuber of varables; s the nuber of rays,.e. the nuber of equatons; M s the nuber of blocs; ter s the nuber of full teratons. ISSN Ýëåêòðîí. ìîäåëèðîâàíèå. 9. Ò. 31. ¹ 49

10 N. Gubaren, M. Pleszczyns a b Fg. 3. The age reconstructon and the absolute error for f x, y) obtaned wth algorth BPART-3 a) and algorth CHBP-3 b) for n =, = 644, M = 36, ter = 7 n the syste 1 1, 1 1) In all experents t was also assued that M s equal to the nuber of detectors; the sequence of chaotc sets I has the for { }, where s an nteger ran varable n the nterval [1, ] wth unfor dstrbuton; the reconstructon an E{ x, y): 1x, y 1} was dvded nto n = pxels; the nuber of projectons n the syste 1 1) s equal to 788, and n the syste 11) the nuber = 644. The results of age reconstructons for f x, y) wth bloc-parallel algorth BPART-3, and chaotc bloc-parallel algorth CHBP-3 n the syste 11, 11) for the sae paraeters are gven n Fg. 3. The plots, whch are presented n Fg. 4, llustrate the dependence of the axu relatve error and the ean absolute error on the nuber of teratons of age reconstructon of f x, y)wth algorths BPART-3 and CHBP-3 n the syste 11, 11). Table 1 shows the dependence of the axu absolute error on the nuber of teratons for algorths BPART-3 and CHBP-3 n the syste 11, 11). The results of reconstructon of the functon f x, y)wth algorth BPART-3 and CHBP-3 n the syste 11, 11) s shown n Fg.. ISSN Electronc Modelng. 9. V. 31. ¹

11 Bloc-Parallel Chaotc Algorths for Iage Reconstructon 1 3 CHBP-3 BPART CHBP-3 BPART ter Fg. 4. Dependence of the ean absolute error and the axu relatve error 1 on the nuber of teratons for age reconstructon of f x, y)wth algorth BPART-3 and CHBP-3 n the syste 1 1, 1 1) The plots presented n Fg. 6 llustrate the dependence of the axu relatve error and the ean absolute error on the nuber of teratons of age reconstructon of f x, y) wth algorths BPART-3 and CHBP-3 n the syste 1 1). Table shows the dependence of the axu absolute error on the nuber of teratons for algorths BPART-3 and CHBP-3 n the syste 1 1). All experental results n the case of reconstructon of objects fro lted projecton data show that the errors of reconstructon wth algorths BPART-3 and CHBP-3 are constantly reduced wth ncreasng the nuber of teratons. Table 1 Iter BPART-3 CHBP Table Iter BPART-3 CHBP ISSN Ýëåêòðîí. ìîäåëèðîâàíèå. 9. Ò. 31. ¹ 1

12 N. Gubaren, M. Pleszczyns a b.4. Fg.. The age reconstructon and the absolute error for f x, y)obtaned wth BPART-3 a) and algorth CHBP-3 b) for n =, = 788, M = 8 n the syste 1 1): a ter = 6; b ter = The obtaned results also show that chaotc algorth CHBP-3 gves better results as copared wth bloc-parallel algorth BPART-3. Table 3 shows the nuber of teratons requred for obtanng the age reconstructon wth a gven axu relatve error for the consdered algorths BPART-3, CHBP-3 and for the consdered systes. Table 3 11, 11) 11) BPART-3 CHBP-3 BPART-3 CHBP-3,% < < < <, ISSN Electronc Modelng. 9. V. 31. ¹

13 Bloc-Parallel Chaotc Algorths for Iage Reconstructon 1 3 CHBP-3 BPART CHBP-3 BPART ter Fg. 6. Dependence of the ean absolute error and the axu relatve error 1 on the nuber of teratons for age reconstructon of f x, y)wth algorth BPART-3 and CHBP-3 n the syste 1 1) All algorths were pleented on IBM/PC processor AMD Duron XP, 16 MHz) by eans of C++ and MATHEMATICA.1. One teraton by eans of Matheatca.1 was pleented approxately 1s for algorth BPART-3 and CHBP-3, and n C++ one teraton for both algorths s pleented n a real te. Concluson. New chaotc teratve algorths for age reconstructon are presented n the paper. These algorths can be realzed on a parallel coputng structure consstng of eleentary processors and soe central processor, all of whch are connected wth shared eory. The qualty and convergence of these algorths were studed by coputng sulaton on sequental coputer. The experental results show that convergent characterstcs of bloc-parallel chaotc algorth CHBP-3 are better as copared wth bloc-parallel algorth BPART-3. Tang nto account that the te of pleentaton of bloc-parallel coputer algorth on parallel coputer s approxately M tes less where M s the nuber of processors) n coparson wth a sequental coputer, t follows fro results of coputer sulaton that the te characterstcs of blocparallel algorths are better as copared wth sequental ART-3. It also follows fro our experents that the confguraton 11, 11) s consderably better as copared wth the schee 11). And for each consdered schee of reconstructon there exst the paraeters whch allow to obtan a rather good qualty of reconstructon after soe nuber of teratons, but ths nuber s consderably larger than that for reconstructon wth coplete projecton data. The nuber of teratons for achevng the stable reconstructon s approxately two tes ore for the second schee n coparson wth the frst one. And ths nuber s approxately tes ore for the schee 11, 11) n coparson wth the case of coplete data. ISSN Ýëåêòðîí. ìîäåëèðîâàíèå. 9. Ò. 31. ¹ 3

14 N. Gubaren, M. Pleszczyns Ðîçðîáëåíî òà âèêîíàíî áëî íî-ïàðàëåëüí³ àëãîðèòìè êîìï þòåðíî òîìîãðàô³. Íàâåäåíî èñåëüí³ àëãîðèòìè â³äíîâëåííÿ òà ðåçóëüòàòè èñåëüíîãî ìîäåëþâàííÿ äëÿ òåñòîâèõ çàäà ³ äåÿêèõ îêðåìèõ âèïàäê³â ñèñòåì ðåêîíñòðóêö³ çáèðàííÿ äàíèõ. 1. Baudet G. M. Asynchronous Iteratve Methods for Multprocessors // J. Assoc. Coput. Mach P Bertseas D. P., Tstsls J. N. Parallel and Dstrbuted Coputaton:Nuercal Methods. Englewood Clffs, NJ : Prentce-Hall, Chazan D., Mraner W. Chaotc Relaxaton//Lnear Alg. ts Appl P Censor Y. Parallel Applcaton of Bloc-teratve Methods n Medcal Iagng and Radaton Therapy // Math. Prograng P De Perro A. R., Iuse A. N. A Sultaneous Projectons Method for Lnear Inequaltes // Lnear Algebra and Its Appl P De Perro A. R., Iuse A. N. A Parallel Projecton Method of Fndng a Coon Pont of a Faly of Convex Sets // Pesqusa Oper P Bru R., Elsner L., Neuann M. Models of Parallel Chaotc Iteraton Methods // Lnear Alg. Its Appl P Gubaren N. Generalzed Model of Asynchronous Iteratons for Iage Reconstructon // Proc. of the Thrd Int. Conf. on PPAM. Kazerz Dolny, Poland, P Gubaren N. Coputer Methods and Algorths for Coputer Toography wth Lted Nuber of Projecton Data. Kev : Nauova Dua, 1997 n Russan).. Gubaren N., Katov A. Sulaton of Parallel Algorths for Coputer Toography // Proc. of the 1-th Europ. Sulaton Multconf. Manchester, Unted Kng, June P Censor Y. Parallel Applcaton of Bloc-teratve Methods n Medcal Iagng and Radaton Therapy // Math. Prograng P De Perro, A. R. Multplcatve Iteratve Methods n Coputer Toography // Lecture Notes n Matheatcs P Gubaren N., Katov A., Szopa J. Parallel Asynchronous Tea Algorth for Iage Reconstructon // Proc. -th IMACS World Congress on Scentfc Coputaton, Modellng and Appled Matheatcs. Berln : Coputatonal Matheatcs ed. by A.Syw). 1997, Vol. 1. P Gubaren N., Pleszczyns M. Chaotc Iteratve Algorths for Iage Reconstructon fro Incoplete Projecton Data // Electronc Modelng. 8. 3, N 3. P Ïîñòóïèëà..9 4 ISSN Electronc Modelng. 9. V. 31. ¹